Help needed with Facial Rigs
I checked your file and the culprit is weight paint. The bones are fine for basics but for good control some constraints and extra control bones should be made.

It is hard to advise about weight paint. That is quite specific to every shape and desired deformations. It is something best learned through training or experience and observing already weighted meshes (for example from open Blender movies).

Instead I am going to give steps to rig a face with Rigify addon, which makes things easier:


Insert a new armature, delete the bone. From Armature properties add the pitchipoy.super_face face template.


Align the face template to the face mesh (scale it to size in edit-mode, not with armature scale). The bones should sit on the surface. Turn on X symmetry while editing so both sides are the same.
From pose or object modes generate the final rig from the template. You can delete the template armature, it has no purpose now:


Bind your mesh to the generated rig with automatic vertex weights. The rig has enough control bones so it should turn out fine. If there are some issues these can be fixed by refining the vertex weights further.
You should now have nice controls and a rig menu with layers in tool shelf.